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Acceleration Due to Gravity

When any object is moved then its speed is changed which is called velocity while the change in velocity is acceleration. When it is changed continues then it is in accelerating condition which is called the rate of acceleration. It is equal to the ratio of change in velocity with respect to time between the given paths. The vector quantity acceleration is used to show the increasing or decreasing speed or changing direction of object.

As we know that a free falling object is under the influence of gravity with an acceleration of 9.8 m/s in downward direction. The free falling objects are free from air resistance. This is the gravitational acceleration or acceleration under gravity denoted with the symbol of g with the standard value of 9.8 m/s but it is varied in different gravitational environments. Gravitational Fields widget is used for investigating the effect of location on the value of g. Let’s discuss about the calculation of acceleration due to gravitational force and its properties.

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What is Acceleration Due to Gravity?

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The most common type of force is the force due to gravity. It is experienced by all of us in every day life. For everyone, falling of an apple from its tree was a common sight and was considered as a natural phenomena. But for Sir Isaac Newton, even though he was a boy at that time, this phenomena made him think inquisitively. Ultimately, he found that the reason of objects falling towards ground is that the Earth has an inherent property of exerting an attraction of force on the objects. The Uniform acceleration Produced in a freely falling body due to gravitational Pull of the earth is Known as acceleration due to gravity.
 
Acceleration of Body = (Force of Body) / (Mass of Body)

                              g = (GM/R^2)

         Where g is the acceleration due to gravity

                     G is the Universal gravitational constant

                     M is the Mass of the Object

                     R is the radius of the Object.
This expression is based on Newton’s second Law of Motion and Law of Universal Gravitation.  

Formulas derived from the fundamental equations of motion.
They are,

v = u + at,

s = ut +$(\frac{1}{2})$at2 and

v2 – u2 = 2as

Where v = Final Velocity

u = Initial Velocity

a = Acceleration

t = time taken.

In case of motion under gravity, the acceleration a and the distance s in the above equations are replaced by gravity g and the height of the object h.

Thus the required equations are,

v = u + gt,

 h = ut + $(\frac{1}{2})$gt2 and

v2 – u2 = 2gh


Where,

h = Height from ground level and

g = acceleration due to gravity.

When an object is thrown vertically upwards with an initial velocity u, the acceleration due to gravity acts as a Negative acceleration. That is, the velocity of the object gets reduced progressively, becomes 0 at a certain height and then the object starts falling like a free fall. The height at which the final velocity becomes 0 is the maximum height that the object can reach for a given initial velocity.

For a vertical throw, formulas can be formed by plugging in v = 0 and g = -g in the fundamental equations of motion. So

t = $\frac{u}{g}$,

h = ut -
$(\frac{1}{2})$gt2 and

u2 = 2gh

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Calculating

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Experimentally

Equations for free fall can be used for calculating acceleration due to gravity. The equation,

h = $(\frac{1}{2})$ gt2 can be solved for g, as,

g = $\frac{2h}{t^{2}}$

Experiment Conducted to find Acceleration due to gravity:
An object of considerable weight (because the air resistance can be neglected), be dropped from fairly tall building whose height is accurately measured. The observer at the top of the building switches on a digital timer at the same instant he drops the object. The observer at ground level switches off the same timer when the object hits the ground and records the time of fall. Plugging the values of h and t, the acceleration due to gravity can be calculated. The experiment may be conducted for a few times and the mean of the readings may be taken.

Theoretically

Bodies allowed to fall freely were found to fall at the same rate irrespective of their masses (air resistance being negligible). The velocity of a freely falling body increased at a steady rate, i.e., the body had acceleration.

We know,

From Newtons Second Law,

F = mg ............ (1)

Equation of Force is given by Newton's law of gravity, which states that every point mass in the universe is attracts to every other point mass in the universe with a force which is directly proportional to the product of their masses and inversely proportional to square of the distance between them. It can be written as,

F = $\frac{GMm}{R^{2}}$ .............(2)

Where

F
is the force,

m
is the mass of the body,

g
is the acceleration due to gravity,

M
is the mass of the Earth,

R
is the radius of the Earth and

G
is the gravitational constant

By knowing these constants, we can calculate "g" theoretically. From equations (1) and (2), we can also conclude that 'g' varies with

(a) Altitude

(b) Depth and

(c) Latitude.

Gravitational Acceleration Constant

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An object acquires acceleration due to gravity because of the force of gravity on that object. There is another type of force of attraction that exists between two objects and the same is called as gravitational force and the acceleration created is called Gravitational acceleration.

Numerically Universal gravitation constant(G) is equal to the value of the forces between two bodies of unit mass separated by a distance of 1 meter.  

The equation of force is stated as "The force of attraction is directly proportional to the product of the masses of the two objects but inversely proportional to the square of the distance between them."

Mathematically, the equation of force in this case is given by,
F = G $\left [ \frac{(m_{1} \times m_{2})}{(d^{2})} \right ]$where,

m1
and m2 are the masses and d is the distance between them.

G
is a constant, known as Gravitational acceleration constant. Its approximate value is 6.674×10-11 N m2 kg-2.
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